Inducing and induced charges -- can be equal or not?

[gracy]

Induced charge can never be greater than inducing charge ,but can it be equal to inducing charge?I think yes.But in my book it is given that "In induction the charge induced in the near surface of a dielectric is lesser and dissimilar,
lesser implies not equal.Why?
I think it is something to do with "dielectric"when body (on which charge is induced)is dielectric, inducing charge can never be greater than inducing charge nor it can be equal.Right?

 

[BvU]

The key here is the term "dielectric". A dielectric does not conduct the charge carriers to where they are most comfortable. Instead, in a dielectric the electric field from a nearby charge causes polarization of the molecules -- meaning they orient themselves so they are more comfortable, but they stay in place. See sheet 54 here (*) and preceding slides. This orientation reduces the field strength, but doesn't bring it all the way down to zero. That corresponds to a net surface charge that is less than the inducing charge. As they say in the sheet:

If the dielectric were replaced with a conductor, the net field between the plates would be zero

zero because the surface charge is then equal to the induced charge.

So the general answer to your question is yes, namely in case of a conductor. But for dielectriccs, which seems to be what the book statement is about, the induced charge is always smaller than the inducing charge.

There is even a quantitative measure for this: the permittivity (that's a pretty hefty link; perhaps hyperphysics is already adequate ... )

[edit] (*) I find that these are sheets from MIT and come from the book
physics for scientists and engineers with modern physics eighth edition by serway and jewett
a 1622 page beauty (with a pdf on the net)

 

[gracy]

the surface charge

?means charge present on Surface of the conductor?

 

[gracy]

zero because the surface charge is then equal to the induced charge.

Here surface charge means charge present on Surface of the conductor?induced charge ?i think there should be inducing charge instead.

 

[BvU]

Keen observer ! Sorry, typo:

zero because the induced surface charge is then equal to the inducing charge.​

 

[gracy]

Keen observer !

I will take it as a compliment.

 

[gracy]

This orientation reduces the field strength, but doesn't bring it all the way down to zero. That corresponds to a net surface charge that is less than the inducing charge.

How electric field gives rise to charge?

 

[BvU]

in a dielectric the electric field from a nearby charge causes polarization of the molecules -- meaning they orient themselves so they are more comfortable

 

[sophiecentaur]

polarization of the molecules

When you are charging an object by induction, the object can be a conductor. This is a much more effective application of induction and will leave the object charged. The whole thing becomes polarised and, by Earthing it, the object plus the Earth become polarised. Disconnecting the earthing wire (or finger) will leave behind the imbalanced charge (on the Earthy end of things). Idealy, I think that the amount of induced charge should be the same as the charge on the originating charged plate (couldn't be more, I think) but there will be a finite amount of Capacitance to other nearby objects and that may well 'dilute' the charge that's induced on the test object.
For an insulator, there is no induced charge left, once it's been removed from the charged source object.
Bits of dust and paper are attracted to a comb because of a low level of induced temporary polarisation (more than induced charge, I should say) but an Electrophorus or Wimshurst Machine will accumulate serious amounts of permanent charge.

 

[gracy]

There is a formula for maximum induced charge
Q'= -Q[1-1/k]
where Q is the inducing charge and Q' is maximum induced charge and k is dielectric constant of material of uncharged body.

 

[gracy]

Q= -Q[1-1/k]

This is equal to
Q'=-Q[k-1/k](negative sign just shows that inducing and induced charge have opposite nature.)
this shows why Q' can never be equal to Q as k-1 can never be equal to k.So k-1/k this ratio will always be less than one i.e zero point something hence induced charge Q' would be always less than inducing charge Q.

 

[sophiecentaur]

There is a formula for maximum induced charge
Q'= -Q[1-1/k]
where Q is the inducing charge and Q' is maximum induced charge and k is dielectric constant of material of uncharged body.

Can you give the context and the experiment that formula refers to? Apart from the effect I of polarisation, I don't understand how a net charge can be Induced (I.e. An opposite charge to the originating object).

 

[gracy]

Can you give the context and the experiment that formula refers to?

I found it in the following video,

 

[sophiecentaur]

Thanks for the picture. I see that I was thinking the other way round! I would like to see a derivation of that formula and see the argument behind it. I can't see where the idea comes from and it would be interesting to find out.

 

[gracy]

If we assume the formula is correct.In that perspective is my post #11 correct?

 

[sophiecentaur]

If we assume the formula is correct.In that perspective is my post #11 correct?

It seems correct that the charge will always less than that value because there will always be a finite distance between the two objects. The k factor must be to do with the potential gradient at the surface (?)

 

[Sisira]

Here, it is said that since (k-1)/k cannot be equal to 1 in the equation Q'= -Q[1-1/k] , induced charge can never be equal to inducing charge. But it is always said in books that induced charge can be equal or less than the inducing charge. How can it be equal then?? Also, whether this is applicable to conductors too?Please reply...